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Minima/ maxima (statement questions) - Number questions
Minima/ maxima (statement questions) - Number questions
Last updated at June 12, 2023 by Teachoo
Ex 6.3, 13 Find two numbers whose sum is 24 and whose product is as large as possible. Let first number be 𝑥 Now, given that First number + Second number = 24 𝑥 + second number = 24 Second number = 24 – 𝑥 Product = (𝑓𝑖𝑟𝑠𝑡 𝑛𝑢𝑚𝑏𝑒𝑟 ) × (𝑠𝑒𝑐𝑜𝑛𝑑 𝑛𝑢𝑚𝑏𝑒𝑟) = 𝑥 (24−𝑥) Let P(𝑥) = 𝑥 (24−𝑥) We need product as large as possible Hence we need to find maximum value of P(𝑥) Finding P’(x) P(𝑥)=𝑥(24−𝑥) P(𝑥)=24𝑥−𝑥^2 P’(𝑥)=24−2𝑥 P’(𝑥)=2(12−𝑥) Putting P’(𝑥)=0 2(12−𝑥)=0 12 – 𝑥 = 0 𝑥 = 12 Finding P’’(𝑥) P’(𝑥)=24−2𝑥 P’’(𝑥) = 0 – 2 = – 2 Thus, p’’(𝑥) < 0 for 𝑥 = 12 𝑥 = 12 is point of maxima & P(𝑥) is maximum at 𝑥 = 12 Finding maximum P(x) P(𝑥)=𝑥(24−𝑥) Putting 𝑥 = 12 p(12)= 12(24−12) = 12(12) = 144 ∴ First number = x = 12 & Second number = 24 – x = (24 – 12)= 12